WO2018233359A1 - Method for recognizing spatial distribution and statistical feature of random dynamic load - Google Patents
Method for recognizing spatial distribution and statistical feature of random dynamic load Download PDFInfo
- Publication number
- WO2018233359A1 WO2018233359A1 PCT/CN2018/083275 CN2018083275W WO2018233359A1 WO 2018233359 A1 WO2018233359 A1 WO 2018233359A1 CN 2018083275 W CN2018083275 W CN 2018083275W WO 2018233359 A1 WO2018233359 A1 WO 2018233359A1
- Authority
- WO
- WIPO (PCT)
- Prior art keywords
- random
- modal
- dynamic load
- random dynamic
- space
- Prior art date
Links
Images
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/15—Correlation function computation including computation of convolution operations
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
Definitions
- the invention relates to a method for identifying spatial distribution of random dynamic loads and statistical features, and belongs to the technical field of structural dynamics inverse problems.
- Dynamic load information on the engineering structure is the basis for structural design and safety assessment. Dynamic load acquisition methods are broadly divided into direct measurement and indirect recognition. In many cases, dynamic loads are difficult to obtain by direct measurement. Generally, the dynamic response of the structure is measured, and the known dynamic load information is identified when the structural system is known.
- the traditional dynamic load identification method uses the structural dynamic response data of a single actual measurement to identify the excitation information that causes the secondary dynamic response, and is a deterministic dynamic load identification method.
- the existing deterministic dynamic load identification method is used to obtain information such as concentrated dynamic load, moving load and distributed dynamic load on the engineering structure. It is worth noting that the distributed dynamic load identification problem is equivalent to identifying an infinite number of concentrated dynamic loads, which is more difficult. Generally, the distributed dynamic load identification problem needs to be reduced in dimension.
- the dynamic loads acting on the actual engineering structure are not only distributed on the structure, but also random.
- the dynamic response will also appear “randomness”; therefore, the structural dynamic response of a single measured measurement can only be one of the samples of the structural random dynamic response information, and the certainty is utilized based on a certain response sample.
- the dynamic load information obtained by the dynamic load identification method can only partially reflect the random dynamic load excitation; in addition, the dynamic response error contained in a single measurement is also used as part of the “true response” in the deterministic dynamic load identification, causing the load. Identify the deviation of the results.
- the traditional deterministic distributed dynamic load identification method and the identification method suitable for centralized random dynamic load can not be used. It is necessary to develop a new method for distributed random dynamic load identification. There are measured structures and dynamic response samples to identify the spatial distribution and statistical characteristics of random dynamic loads.
- the object of the present invention is to provide a method for identifying the spatial distribution of random dynamic loads and the identification of statistical features, and solving the problem of statistical characteristics and distribution characteristics of random dynamic loads on the structure of the dynamic response sample identification structure in the time domain, for serving in distributed randomization
- the engineering structure design and safety assessment under dynamic load environment provides accurate and reliable dynamic load information.
- a method for identifying a spatial distribution of random dynamic loads and a statistical feature characterized in that the method comprises the following steps:
- the structural random vibration response is developed by using the mode shape to obtain the dynamic response of the structure in the modal space;
- the rth measurement acquisition displacement vector sample W r at the position of the structure (x 1 , x 2 , ... x n ) is expressed as:
- w r (x j , t) represents the value of the structural displacement obtained at the rth measurement at time t at x j
- N is the number of measurements, and may also be considered as the response sample size
- the modal shape function is used to calculate the modal displacement vector corresponding to the rth measurement in the modal space:
- the random dynamic load distribution is independent of its random characteristics and time history, and its distribution function and random time history are all developed by using the mode shape.
- the specific steps are as follows: the distributed random dynamic load f(x, t, ⁇ ) is expressed.
- the product of its distribution function T(x) and the random time history P(t, ⁇ ) is as follows:
- the distribution function T(x) is expanded using the modal shape:
- the random dynamic load f i (t, ⁇ ) in the i-th modal space can be expressed as:
- L represents the length of the beam
- S51 The spatial distribution of the random dynamic load can be calculated by:
- t 1 is any time
- the invention has the following advantages:
- the existing random dynamic load identification technology can only identify the random concentrated dynamic load on the structure by the measured structure dynamic response sample.
- the existing distributed random dynamic load identification method cannot be applied to the identification of non-stationary random dynamic load.
- the distributed random dynamic load time domain identification technology provided by the invention can utilize the measured structure dynamic response sample at the limited measurement point to identify the spatial distribution of the random dynamic load and the variation of the statistical characteristics with time, and has certain advancement;
- the problem of identifying the distributed random dynamic load is transformed into the estimation problem of the dynamic process of the dynamic load in the modal space, which greatly reduces the dimension and difficulty of the load identification problem, and has easy operation and computational efficiency. High features.
- Figure 1 is a logic flow diagram of the method of the present invention.
- Figure 2 is a schematic diagram of a simply supported beam under distributed random dynamic loads.
- Figure 3 is a schematic diagram showing the results of spatial distribution identification of random dynamic loads.
- Fig. 4(a) is a schematic diagram showing the results of time-varying mean value identification of random dynamic loads.
- Fig. 4(b) is a schematic diagram showing the results of time-varying variance identification of random dynamic loads.
- Figure 4(c) is a schematic diagram showing the results of the random dynamic load correlation function.
- Embodiment The distributed random dynamic load acting on the simply supported beam structure as shown in Fig. 2 is identified by the method of the present invention.
- the trapezoidal distributed random dynamic load distribution function to be identified is:
- the stochastic dynamic load component F(t, ⁇ ) of distributed random dynamic load is divided into two parts: deterministic dynamic load and random dynamic load.
- the spatial distribution and statistical characteristics of the random dynamic load are identified by the measured dynamic random response sample by using the technique of the present invention, and specifically includes the following steps:
- S1 Perform modal test on the simply supported beam, and obtain the first five natural frequencies of the structure are 3.9 Hz, 15.6 Hz, 35.1 Hz, 62.5 Hz and 97.6 Hz, respectively, and obtain the mode shapes corresponding to the natural frequencies of each order;
- the rth measurement acquisition displacement response sample vector W r at each position on the beam structure is expressed as:
- w r (x j , t) represents the value of the structural displacement obtained at the rth measurement at time t at x j
- N is the number of measurements, which is 5000 times in this embodiment, that is, the total number of samples is 5000.
- the modal shape function is used to calculate the modal displacement vector corresponding to the rth measurement in the modal space:
- q i,r (t) is the modal displacement of the structural displacement obtained in the rth measurement in the i-th modal space
- the upper right corner + sign indicates the generalized inverse.
- S31 The random dynamic load distribution is independent of its random characteristics and time history, and its distribution function and random time history are all developed by using the mode shape. Specific steps are as follows:
- the random dynamic load f i (t, ⁇ ) in the i-th modal space can be expressed as:
- L represents the length of the beam.
- ⁇ A is the linear density of the beam.
- the identification method in the present invention can accurately identify the statistical characteristics of the random dynamic load with time by using the response samples at the limited measurement points, and is suitable for the case of non-stationary random dynamic loads, and has certain advancement.
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Mathematical Physics (AREA)
- Data Mining & Analysis (AREA)
- Mathematical Optimization (AREA)
- Computational Mathematics (AREA)
- Pure & Applied Mathematics (AREA)
- Mathematical Analysis (AREA)
- General Engineering & Computer Science (AREA)
- Software Systems (AREA)
- Databases & Information Systems (AREA)
- Algebra (AREA)
- Operations Research (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Geometry (AREA)
- Computing Systems (AREA)
- Measurement Of Mechanical Vibrations Or Ultrasonic Waves (AREA)
- Testing Of Devices, Machine Parts, Or Other Structures Thereof (AREA)
Abstract
A method for recognizing the spatial distribution and statistical features of a random dynamic load. The method comprises the steps of: S1. carrying out a modal experiment so as to acquire mode parameters of a structure, the parameters comprising an inherent frequency and a mode shape; S2. expanding a random vibration response of the structure by using the mode shape so as to acquire a dynamic response, in a mode space, of the structure; S3. by means of mode shape expansion, projecting random dynamic loads, distributed with the space, onto the mode space; S4. in the mode space, inverting random dynamic load samples according to random dynamic response samples; and S5. calculating the spatial distribution and statistical features of a random dynamic load on the structure according to the random dynamic load samples in the mode space and mode shape functions. The method can solve the problem of recognizing, in a time domain, the statistical properties and distribution features of random dynamic loads on a structure by means of actually measured dynamic response samples of the structure, and provide accurate and reliable dynamic load information for the design and safety assessment of an engineering structure serving in a distributed random dynamic load environment.
Description
本发明涉及一种随机动载荷空间分布及统计特征的识别方法，属于结构动力学反问题技术领域。The invention relates to a method for identifying spatial distribution of random dynamic loads and statistical features, and belongs to the technical field of structural dynamics inverse problems.
工程结构上的动载荷信息是结构设计和安全评估的依据。动载荷的获取方法大体分为直接测量和间接识别两种。在许多情况下，动载荷难以通过直接测量获得，一般采用测量结构上的动响应，在结构系统已知的情况下，识别获得所需动载荷信息。Dynamic load information on the engineering structure is the basis for structural design and safety assessment. Dynamic load acquisition methods are broadly divided into direct measurement and indirect recognition. In many cases, dynamic loads are difficult to obtain by direct measurement. Generally, the dynamic response of the structure is measured, and the known dynamic load information is identified when the structural system is known.
传统的动载荷识别方法是利用单次实测的结构动响应数据识别引起该次动响应的激励信息，是确定性动载荷识别方法。现有的确定性动载荷识别方法被用于获取工程结构上的集中动载荷，移动载荷以及分布式动载荷等信息。值得注意的是，分布式动载荷识别问题相当于识别无穷多个集中动载荷，难度更大，一般需要将分布式动载荷识别问题降维求解。The traditional dynamic load identification method uses the structural dynamic response data of a single actual measurement to identify the excitation information that causes the secondary dynamic response, and is a deterministic dynamic load identification method. The existing deterministic dynamic load identification method is used to obtain information such as concentrated dynamic load, moving load and distributed dynamic load on the engineering structure. It is worth noting that the distributed dynamic load identification problem is equivalent to identifying an infinite number of concentrated dynamic loads, which is more difficult. Generally, the distributed dynamic load identification problem needs to be reduced in dimension.
实际工程结构上作用的动载荷，如建筑物上的风载荷，海洋平台承受的海浪载荷以及飞行器表面的气动载荷等，不仅分布于结构上，而且还具有随机性。随机动载荷施加于结构时，动响应也将随之呈现“随机性”；因此，单次实测的结构动响应只能是结构随机动响应信息的其中一个样本，基于某个响应样本利用确定性动载荷识别方法获得的动载荷信息也只能部分反映该随机动载荷激励；另外，单次测量中包含的动响应误差在确定性动载荷识别中也被作为“真实响应”的一部分，引起载荷识别结果的偏差。针对此类分布式随机动载荷的识别问题，传统的确定性分布动载荷识别方法和适用于集中随机动载荷的识别方法均无法使用，需要发展一种针对分布式随机动载荷识别的新方法，有实测结构动响应样本识别随机动载荷的空间分布和统计特征。The dynamic loads acting on the actual engineering structure, such as the wind load on the building, the wave load on the ocean platform and the aerodynamic load on the surface of the aircraft, are not only distributed on the structure, but also random. When a random dynamic load is applied to the structure, the dynamic response will also appear “randomness”; therefore, the structural dynamic response of a single measured measurement can only be one of the samples of the structural random dynamic response information, and the certainty is utilized based on a certain response sample. The dynamic load information obtained by the dynamic load identification method can only partially reflect the random dynamic load excitation; in addition, the dynamic response error contained in a single measurement is also used as part of the “true response” in the deterministic dynamic load identification, causing the load. Identify the deviation of the results. For the identification of such distributed random dynamic loads, the traditional deterministic distributed dynamic load identification method and the identification method suitable for centralized random dynamic load can not be used. It is necessary to develop a new method for distributed random dynamic load identification. There are measured structures and dynamic response samples to identify the spatial distribution and statistical characteristics of random dynamic loads.
发明内容Summary of the invention
本发明的目的是提供一种随机动载荷空间分布及统计特征的识别方法，解决在时域内利用实测结构动响应样本识别结构上随机动载荷的统计特性和分布特征问题，为服役于分布式随机动载荷环境下的工程结构设计与安全评估提供准确可靠的动载荷信息。The object of the present invention is to provide a method for identifying the spatial distribution of random dynamic loads and the identification of statistical features, and solving the problem of statistical characteristics and distribution characteristics of random dynamic loads on the structure of the dynamic response sample identification structure in the time domain, for serving in distributed randomization The engineering structure design and safety assessment under dynamic load environment provides accurate and reliable dynamic load information.
上述的目的通过以下技术方案实现：The above objectives are achieved by the following technical solutions:
一种随机动载荷空间分布及统计特征的识别方法，其特征在于，该方法包括如下步骤：A method for identifying a spatial distribution of random dynamic loads and a statistical feature, characterized in that the method comprises the following steps:
S1.开展模态试验，获取结构的模态参数，包括固有频率和模态振型；S1. Carry out a modal test to obtain the modal parameters of the structure, including the natural frequency and the mode shape;
S2.将结构随机振动响应利用模态振型展开，获取结构在模态空间的动响应；S2. The structural random vibration response is developed by using the mode shape to obtain the dynamic response of the structure in the modal space;
S3.利用模态振型展开，将随空间分布的随机动载荷投影到模态空间；S3. Using a mode shape expansion, projecting a random dynamic load distributed with space into a modal space;
S4.在模态空间内，由随机动响应样本反演随机动载荷样本；S4. Inverting the random dynamic load sample from the random motion response sample in the modal space;
S5.由模态空间内随机动载荷样本及振型函数求解结构上随机动载荷的空间分布和统计特征。S5. Solving the spatial distribution and statistical characteristics of random dynamic loads on a structure from random dynamic load samples and mode functions in modal space.
2.根据权利要求1所述的随机动载荷空间分布及统计特征的识别方法，其特征在于，步骤S2中所述的将结构随机振动响应利用模态振型展开，获取结构在模态空间的动响应，具体步骤为：2 . The method for identifying a spatial distribution of random dynamic loads and a statistical feature according to claim 1 , wherein the random vibration response of the structure is expanded by using a mode shape to obtain a structure in a modal space. Dynamic response, the specific steps are:
S21：利用多次重复测量方式获取随机振动响应的样本集合：S21: Acquiring a sample set of random vibration responses by using multiple repeated measurements:
在结构上(x
1,x
2,…x
n)位置处第r次测量获取位移响应样本向量W
r表示为：
The rth measurement acquisition displacement vector sample W r at the position of the structure (x 1 , x 2 , ... x n ) is expressed as:
W
r＝{w
r(x
1,t) w
r(x
2,t) … w
r(x
n,t)}
T,r＝1,…,N (1)，
W r ={w r (x 1 ,t) w r (x 2 ,t) ... w r (x n ,t)} T , r=1,...,N (1),
其中w
r(x
j,t)表示第r次测量获得的结构位移在x
j处t时刻的值，N为测量的次数，也可认为响应样本容量；
Where w r (x j , t) represents the value of the structural displacement obtained at the rth measurement at time t at x j , and N is the number of measurements, and may also be considered as the response sample size;
S22：针对单个样本，即单次实测结构振动响应，利用模态振型展开获得结构振动在模态空间内响应：S22: For a single sample, that is, a single measured structural vibration response, the modal vibration mode is used to obtain the structural vibration response in the modal space:
利用模态振型函数计算模态空间内第r次测量对应的模态位移向量：The modal shape function is used to calculate the modal displacement vector corresponding to the rth measurement in the modal space:
其中q
i,r(t)为第r次测量获得的结构位移在第i阶模态空间中的模态位移，
表示第i阶模态振型函数在x
j处的值，右上角+号表示广义逆。
Where q i,r (t) is the modal displacement of the structural displacement obtained in the rth measurement in the i-th modal space, Indicates the value of the i-th mode mode shape function at x j , and the upper right corner + sign indicates the generalized inverse.
3.根据权利要求1所述的随机动载荷空间分布及统计特征的识别方法，其特征在于，步骤S3中所述的利用模态振型展开，将随空间分布的随机动载荷投影到模态空间，具体包括以下步骤：The method for identifying a spatial distribution of random dynamic loads and a statistical feature according to claim 1, wherein the modal vibration mode expansion is performed in step S3, and a random dynamic load distributed with space is projected to a modal state. Space, specifically including the following steps:
S31：令随机动载荷分布独立于其随机特性和时间历程，将其分布函数和随机时间历程均利用模态振型展开，具体步骤如下：将分布随机动载荷f(x,t,θ)表示为其分布函数T(x)与随机时间历程P(t,θ)的乘积，如下式所示：S31: The random dynamic load distribution is independent of its random characteristics and time history, and its distribution function and random time history are all developed by using the mode shape. The specific steps are as follows: the distributed random dynamic load f(x, t, θ) is expressed. The product of its distribution function T(x) and the random time history P(t, θ) is as follows:
f(x,t,θ)＝T(x)·P(t,θ) (3)，f(x,t,θ)=T(x)·P(t,θ) (3),
将分布函数T(x)利用模态振型展开：The distribution function T(x) is expanded using the modal shape:
将式(4)代入式(3)，可得：Substituting equation (4) into equation (3), you can get:
其中a
k(t,θ)＝d
kP(t,θ)。
Where a k (t, θ) = d k P(t, θ).
第i阶模态空间内的随机动载荷f
i(t,θ)可以表示为：
The random dynamic load f i (t, θ) in the i-th modal space can be expressed as:
其中L表示梁的长度；Where L represents the length of the beam;
S32：利用模态振型函数的正交性，建立模态空间内随机动响应与随机动载荷之间的关系。具体步骤如下：S32: Using the orthogonality of the mode shape function, establishing a relationship between the random dynamic response and the random dynamic load in the modal space. Specific steps are as follows:
结构在模态空间内的动力学方程为：The dynamic equations of the structure in the modal space are:
其中q
i(t,θ)，
和
分别为模态空间内的结构随机位移，速度和加速度，ζ
i和m
i分别为第i阶模态阻尼比和模态质量。将式(6)代入式(7)，利用结构模态振型的正交性条件，可以获得模态空间内随机动响应与随机动载荷之间的关系式，如下：
Where q i (t, θ), with The random displacement, velocity and acceleration of the structure in the modal space, ζ i and m i are the i-th modal damping ratio and modal mass, respectively. Substituting equation (6) into equation (7), using the orthogonality condition of the structural mode shape, the relationship between the random dynamic response and the random dynamic load in the modal space can be obtained as follows:
4.根据权利要求1所述的随机动载荷空间分布及统计特征的识别方法，其特征在于，步骤S4中所述的在模态空间内，由随机动响应求解随机动载荷，具体包括以下步骤：The method for identifying a random dynamic load spatial distribution and a statistical feature according to claim 1, wherein in the modal space, the random dynamic response is solved by the random dynamic response in the modal space, and the following steps are specifically included. :
S41：根据结构位移，速度和加速度之间的导数关系，由第r次测量获得的结构模态位移q
i,r(t)，对时间求导获取对应的模态速度和模态加速度；若测量获得的是结构加速度信号，同样可采用积分的方式获取速度和位移；
S41: According to the structural displacement, the derivative relationship between the velocity and the acceleration, the structural modal displacement q i,r (t) obtained by the rth measurement, and the time derivative to obtain the corresponding modal velocity and modal acceleration; The measurement obtains the structural acceleration signal, and the speed and displacement can also be obtained by integrating;
S42：由式(8)中的模态空间内随机位移q
i(t,θ)的样本q
i,r(t)及其导数，求解得到模态空间内随机动载荷a
i(t,θ)的样本a
i,r(t)。
S42: Calculating the random dynamic load a i (t, θ in the modal space by the sample q i,r (t) and its derivative of the random displacement q i (t, θ) in the modal space in the equation (8) The sample a i,r (t).
5.根据权利要求1所述的随机动载荷空间分布及统计特征的识别方法，其特征在于，步骤S5中所述的由模态空间内随机动载荷样本及振型函数求解结构上随机动载荷的空间分布和统计特征，具体包括以下步骤：The method for identifying spatial distribution and statistical features of random dynamic loads according to claim 1, wherein the random dynamic load samples and the mode shape function in the modal space are used to solve the random dynamic load on the structure described in step S5. Spatial distribution and statistical characteristics, including the following steps:
S51：随机动载荷的空间分布可以由下式计算：S51: The spatial distribution of the random dynamic load can be calculated by:
其中t
1为任意时刻；
Where t 1 is any time;
S52：随机动载荷的均值μ
f(x,t)可以由下式计算：
S52: The mean value of the random dynamic load μ f (x, t) can be calculated by:
S53：随机动载荷的方差Var
f(x,t)可以由下式计算：
S53: The variance of the random dynamic load Var f (x, t) can be calculated by:
本发明与现有技术相比，具有以下优点：Compared with the prior art, the invention has the following advantages:
1、现有的随机动载荷识别技术一般只能由实测结构动响应样本识别结构上随机集中动载荷，目前已经出现的分布随机动载荷识别方法无法适用于非平稳随机动载荷的识别，而本发明中提供的分布随机动载荷时域识别技术能够利用有限测点处的实测结构动响应样本识别随机动载荷的空间分布和统计特征随时间的变化规律，具有一定的先进性；1. The existing random dynamic load identification technology can only identify the random concentrated dynamic load on the structure by the measured structure dynamic response sample. The existing distributed random dynamic load identification method cannot be applied to the identification of non-stationary random dynamic load. The distributed random dynamic load time domain identification technology provided by the invention can utilize the measured structure dynamic response sample at the limited measurement point to identify the spatial distribution of the random dynamic load and the variation of the statistical characteristics with time, and has certain advancement;
2、利用模态振型函数展开，将分布随机动载荷的识别问题转换为模态空间内动载荷随机过程的估计问题，大大降低了载荷识别问题的维数和难度，具有易操作和计算效率高的特点。2. Using the mode shape function expansion, the problem of identifying the distributed random dynamic load is transformed into the estimation problem of the dynamic process of the dynamic load in the modal space, which greatly reduces the dimension and difficulty of the load identification problem, and has easy operation and computational efficiency. High features.
图1为本发明方法的逻辑流程框图。Figure 1 is a logic flow diagram of the method of the present invention.
图2为分布随机动载荷作用下简支梁示意图。Figure 2 is a schematic diagram of a simply supported beam under distributed random dynamic loads.
图3为随机动载荷空间分布识别结果示意图。Figure 3 is a schematic diagram showing the results of spatial distribution identification of random dynamic loads.
图4(a)为随机动载荷时变均值识别结果示意图。Fig. 4(a) is a schematic diagram showing the results of time-varying mean value identification of random dynamic loads.
图4(b)为随机动载荷时变方差识别结果示意图。Fig. 4(b) is a schematic diagram showing the results of time-varying variance identification of random dynamic loads.
图4(c)为随机动载荷相关函数识别结果示意图。Figure 4(c) is a schematic diagram showing the results of the random dynamic load correlation function.
下面通过实施例的方式，对本发明技术方案进行详细说明，但实施例仅是本发明的其中一种实施方式，应当指出：对于本技术领域的技术人员来说，在不脱离本发明原理的前提下，还可以以改变结构和载荷形式等方式做出若干改进和等同替换，这些对本发明权利要求进行改进和等同替换后的技术方案，均落入本发明的保护范围。The technical solutions of the present invention are described in detail below by means of the embodiments, but the embodiments are only one of the embodiments of the present invention, and it should be noted that those skilled in the art can not deviate from the principles of the present invention. In the following, it is also possible to make a number of modifications and equivalent substitutions in a manner that changes the structure and the form of the load, and the technical solutions of the present invention are modified and equivalently substituted.
实施例：对如图2所示简支梁结构上作用的分布随机动载荷利用本发明的方法进行识别。梁长L＝40m，横截面积A＝4.8m
2，截面惯性矩I＝2.5498m
4，结构的阻尼采用瑞利阻尼，各阶模态阻尼比ξ
i＝0.02，材料的弹性模量E＝5×10
10N/m
2，密度ρ＝2.5×10
3kg/m
3。待识别的梯形分 布随机动载荷分布函数为：
Embodiment: The distributed random dynamic load acting on the simply supported beam structure as shown in Fig. 2 is identified by the method of the present invention. Beam length L=40m, cross-sectional area A=4.8m 2 , section moment of inertia I=2.5498m 4 , the damping of the structure adopts Rayleigh damping, the modal damping ratio of each order is ξ i =0.02, the elastic modulus of the material E= 5 × 10 10 N/m 2 , density ρ = 2.5 × 10 3 kg / m 3 . The trapezoidal distributed random dynamic load distribution function to be identified is:
分布式随机动载荷的随机性动载荷组分F(t,θ)分为确定性动载荷和随机性动载荷两个部分。The stochastic dynamic load component F(t, θ) of distributed random dynamic load is divided into two parts: deterministic dynamic load and random dynamic load.
F(t,θ)的确定性动载荷部分：The deterministic dynamic load portion of F(t, θ):
F
d(t)＝20000[1+0.1sin(2πt)]N (2)
F d (t)=20000[1+0.1sin(2πt)]N (2)
F(t,θ)的随机性动载荷部分假定为零均值非平稳高斯随机过程，功率谱函数S(ω,t)为：The random dynamic load portion of F(t, θ) assumes a zero-mean non-stationary Gaussian random process, and the power spectrum function S(ω, t) is:
S(ω,t)＝C
fP
d(t)Φ(ω) (3)
S(ω,t)=C f P d (t)Φ(ω) (3)
其中：C
f表示随机水平，取C
f＝0.2；Φ(ω)表示零均值非平稳高斯随机过程的功率谱密度函数，有Φ(ω)＝(1/2π)(2/ω
2+1)。
Where: C f represents the random level, taking C f =0.2; Φ(ω) represents the power spectral density function of the zero-mean non-stationary Gaussian random process, with Φ(ω)=(1/2π)(2/ω 2 +1 ).
利用本发明的技术由实测结构随机动响应样本识别随机动载荷的空间分布和统计特征，具体包括以下步骤：The spatial distribution and statistical characteristics of the random dynamic load are identified by the measured dynamic random response sample by using the technique of the present invention, and specifically includes the following steps:
S1：对简支梁开展模态试验，获取结构的前五阶固有频率分别为3.9Hz,15.6Hz,35.1Hz,62.5Hz和97.6Hz，同时获得各阶固有频率所对应的模态振型；S1: Perform modal test on the simply supported beam, and obtain the first five natural frequencies of the structure are 3.9 Hz, 15.6 Hz, 35.1 Hz, 62.5 Hz and 97.6 Hz, respectively, and obtain the mode shapes corresponding to the natural frequencies of each order;
S2：将结构随机振动响应利用模态振型展开，利用多次测量获取位移响应信号，求解结构在模态空间的动响应，包括以下步骤：S2: The structural random vibration response is developed by using a mode shape, and the displacement response signal is obtained by using multiple measurements to solve the dynamic response of the structure in the modal space, including the following steps:
S21：在梁结构上均匀布置19个测点，重复测量各测点处的动态位移信号，获取在分布随机激励下随机振动响应的样本集合；S21: uniformly arranging 19 measuring points on the beam structure, repeatedly measuring the dynamic displacement signals at each measuring point, and obtaining a sample set of random vibration responses under distributed random excitation;
S22：针对单个样本，即单次实测结构振动响应，利用模态振型展开获得结构振动在模态空间内响应。具体步骤如下：S22: For a single sample, that is, a single measured structural vibration response, the modal vibration mode is used to obtain the structural vibration response in the modal space. Specific steps are as follows:
梁结构上各位置处第r次测量获取位移响应样本向量W
r表示为：
The rth measurement acquisition displacement response sample vector W r at each position on the beam structure is expressed as:
W
r＝{w
r(x
1,t) w
r(x
2,t) … w
r(x
n,t)}
T,r＝1,…,N (4)，
W r ={w r (x 1 ,t) w r (x 2 ,t) ... w r (x n ,t)} T , r=1,...,N (4),
其中w
r(x
j,t)表示第r次测量获得的结构位移在x
j处t时刻的值，N为测量的次数，本实施例中采用5000次，即为样本总数为5000。利用模态振型函数计算模态空间内第r次测量对应的模态位移向量：
Where w r (x j , t) represents the value of the structural displacement obtained at the rth measurement at time t at x j , and N is the number of measurements, which is 5000 times in this embodiment, that is, the total number of samples is 5000. The modal shape function is used to calculate the modal displacement vector corresponding to the rth measurement in the modal space:
其中q
i,r(t)为第r次测量获得的结构位移在第i阶模态空间中的模态位移，
表示第i 阶模态振型函数在x
j处的值，右上角+号表示广义逆。此时，测点数n＝19，模态数m＝5。
Where q i,r (t) is the modal displacement of the structural displacement obtained in the rth measurement in the i-th modal space, Indicates the value of the i-th mode mode shape function at x j , and the upper right corner + sign indicates the generalized inverse. At this time, the number of measurement points is n=19, and the number of modes is m=5.
S3：利用模态振型展开，将随空间分布的随机动载荷投影到模态空间，包括以下步骤：S3: Using the mode shape expansion, projecting the random dynamic load distributed along the space into the modal space, including the following steps:
S31：令随机动载荷分布独立于其随机特性和时间历程，将其分布函数和随机时间历程均利用模态振型展开。具体步骤如下：S31: The random dynamic load distribution is independent of its random characteristics and time history, and its distribution function and random time history are all developed by using the mode shape. Specific steps are as follows:
以一梯形分布的非平稳随机动载荷f(x,t,θ)为例，可以表示为其分布函数T(x)与随机时间历程P(t,θ)的乘积，如下式所示：Taking a trapezoidal distributed non-stationary random dynamic load f(x, t, θ) as an example, it can be expressed as the product of its distribution function T(x) and the random time history P(t, θ), as shown in the following equation:
f(x,t,θ)＝T(x)·P(t,θ) (6)，f(x,t,θ)=T(x)·P(t,θ) (6),
根据下式利用前5阶固有振型函数展开分布函数T(x)，即k＝5：The distribution function T(x) is expanded by the first five-order natural mode function according to the following equation, that is, k=5:
利用(7)式中的振型展开，识别动载荷的分布函数即可转换为识别系数d
k的问题，大大降低识别难度。将式(7)代入式(6)，可得
By using the mode expansion in (7), the distribution function of the dynamic load can be recognized as a problem of the recognition coefficient d k , which greatly reduces the difficulty of recognition. Substituting equation (7) into equation (6),
其中a
k(t,θ)＝d
kP(t,θ)。
Where a k (t, θ) = d k P(t, θ).
第i阶模态空间内的随机动载荷f
i(t,θ)可以表示为：
The random dynamic load f i (t, θ) in the i-th modal space can be expressed as:
其中L表示梁的长度。Where L represents the length of the beam.
S32：利用模态振型函数的正交性，建立模态空间内随机动响应与随机动载荷之间的关系。具体步骤如下：S32: Using the orthogonality of the mode shape function, establishing a relationship between the random dynamic response and the random dynamic load in the modal space. Specific steps are as follows:
简支梁在模态空间内的动力学方程为：The dynamic equation of the simply supported beam in the modal space is:
其中q
i(t,θ)，
和
分别为模态空间内的结构随机位移，速度和加速度，ζ
i和m
i分别为第i阶模态阻尼比和模态质量。将式(9)代入式(10)，利用简支梁模态振型的正交性条件：
Where q i (t, θ), with The random displacement, velocity and acceleration of the structure in the modal space, ζ i and m i are the i-th modal damping ratio and modal mass, respectively. Substituting equation (9) into equation (10), using the orthogonality condition of the mode of the simply supported beam mode:
其中δ
ij为克罗内克函数。则可以获得模态空间内随机动响应与随机动载荷之间的关系式，如下：
Where δ ij is the Kroneck function. The relationship between the random dynamic response and the random dynamic load in the modal space can be obtained as follows:
其中ρA为梁的线密度。Where ρA is the linear density of the beam.
S4：在模态空间内，由随机动响应求解随机动载荷。具体步骤如下：S4: In the modal space, the random dynamic load is solved by the random dynamic response. Specific steps are as follows:
S41：根据结构位移，速度和加速度之间的导数关系，由第r次测量获得的结构模态位移q
i,r(t)，对时间求导获取对应的模态速度和模态加速度；若测量获得的是结构加速度信号，同样可采用积分的方式获取速度和位移；
S41: According to the structural displacement, the derivative relationship between the velocity and the acceleration, the structural modal displacement q i,r (t) obtained by the rth measurement, and the time derivative to obtain the corresponding modal velocity and modal acceleration; The measurement obtains the structural acceleration signal, and the speed and displacement can also be obtained by integrating;
S42：由式(12)中的模态空间内随机位移q
i(t,θ)的样本q
i,r(t)及其导数，求解得到模态空间内随机动载荷a
i(t,θ)的样本a
i,r(t)。
S42: Calculating the random dynamic load a i (t, θ in the modal space by the sample q i,r (t) and its derivative of the random displacement q i (t, θ) in the modal space in the equation (12) The sample a i,r (t).
S5：由模态空间内随机动载荷及振型函数求解结构上随机动载荷的空间分布和统计特征。具体步骤如下：S5: Solving the spatial distribution and statistical characteristics of the random dynamic load on the structure by the random dynamic load and the mode function in the modal space. Specific steps are as follows:
S51：利用所提出方法，识别获得各个时刻随机动载荷空间分布，与真实分布对比如图3所示；S51: using the proposed method, identifying and obtaining the spatial distribution of the random dynamic load at each moment, and comparing with the real distribution is shown in FIG. 3;
S52：识别获得随机动载荷随时间变化的均值，梁跨中处随机动载荷识别均值与真实值对比如图4所示；S52: Identifying the mean value of the random dynamic load over time, and comparing the mean value of the random dynamic load identification with the true value in the beam span as shown in FIG. 4;
S53：识别获得随机动载荷随时间变化的方差和相关函数，梁跨中处随机动载荷方差和相关函数识别值与真实值对比如图4所示；S53: Identifying the variance and correlation function of the random dynamic load over time, and comparing the variance of the random dynamic load and the correlation function between the beam span and the true value as shown in FIG. 4;
由上可知，本发明中的识别方法能够利用有限测点处响应样本准确识别随机动载荷随时间变化的统计特征，适用于非平稳随机动载荷的情况，具有一定先进性。It can be seen from the above that the identification method in the present invention can accurately identify the statistical characteristics of the random dynamic load with time by using the response samples at the limited measurement points, and is suitable for the case of non-stationary random dynamic loads, and has certain advancement.
Claims (5)
- 一种随机动载荷空间分布及统计特征的识别方法，其特征在于，该方法包括如下步骤：A method for identifying a spatial distribution of random dynamic loads and a statistical feature, characterized in that the method comprises the following steps:S1.开展模态试验，获取结构的模态参数，包括固有频率和模态振型；S1. Carry out a modal test to obtain the modal parameters of the structure, including the natural frequency and the mode shape;S2.将结构随机振动响应利用模态振型展开，获取结构在模态空间的动响应；S2. The structural random vibration response is developed by using the mode shape to obtain the dynamic response of the structure in the modal space;S3.利用模态振型展开，将随空间分布的随机动载荷投影到模态空间；S3. Using a mode shape expansion, projecting a random dynamic load distributed with space into a modal space;S4.在模态空间内，由随机动响应样本反演随机动载荷样本；S4. Inverting the random dynamic load sample from the random motion response sample in the modal space;S5.由模态空间内随机动载荷样本及振型函数求解结构上随机动载荷的空间分布和统计特征。S5. Solving the spatial distribution and statistical characteristics of random dynamic loads on a structure from random dynamic load samples and mode functions in modal space.
- 根据权利要求1所述的随机动载荷空间分布及统计特征的识别方法，其特征在于，步骤S2中所述的将结构随机振动响应利用模态振型展开，获取结构在模态空间的动响应，具体步骤为：The method for identifying a spatial distribution of random dynamic loads and a statistical feature according to claim 1, wherein the random vibration response of the structure is expanded by using a mode shape to obtain a dynamic response of the structure in a modal space. The specific steps are:S21：利用多次重复测量方式获取随机振动响应的样本集合：S21: Acquiring a sample set of random vibration responses by using multiple repeated measurements:在结构上(x 1,x 2,…x n)位置处第r次测量获取位移响应样本向量W r表示为： The rth measurement acquisition displacement vector sample W r at the position of the structure (x 1 , x 2 , ... x n ) is expressed as:W r＝{w r(x 1,t) w r(x 2,t) … w r(x n,t)} T,r＝1,…,N (1)， W r ={w r (x 1 ,t) w r (x 2 ,t) ... w r (x n ,t)} T , r=1,...,N (1),其中w r(x j,t)表示第r次测量获得的结构位移在x j处t时刻的值，N为测量的次数，也可认为响应样本容量； Where w r (x j , t) represents the value of the structural displacement obtained at the rth measurement at time t at x j , and N is the number of measurements, and may also be considered as the response sample size;S22：针对单个样本，即单次实测结构振动响应，利用模态振型展开获得结构振动在模态空间内响应：S22: For a single sample, that is, a single measured structural vibration response, the modal vibration mode is used to obtain the structural vibration response in the modal space:利用模态振型函数计算模态空间内第r次测量对应的模态位移向量：The modal shape function is used to calculate the modal displacement vector corresponding to the rth measurement in the modal space:
其中q i,r(t)为第r次测量获得的结构位移在第i阶模态空间中的模态位移，表示第i阶模态振型函数在x j处的值，右上角+号表示广义逆。 Where q i,r (t) is the modal displacement of the structural displacement obtained in the rth measurement in the i-th modal space,
Indicates the value of the i-th mode mode shape function at x j , and the upper right corner + sign indicates the generalized inverse. - 根据权利要求1所述的随机动载荷空间分布及统计特征的识别方法，其特征在于，步骤S3中所述的利用模态振型展开，将随空间分布的随机动载荷投影到模态空间，具体包括以下步骤：The method for identifying a spatial distribution of random dynamic loads and a statistical feature according to claim 1, wherein the modal vibration mode expansion is performed in step S3, and a random dynamic load distributed with space is projected into the modal space. Specifically, the following steps are included:S31：令随机动载荷分布独立于其随机特性和时间历程，将其分布函数和随机时间历程均利用模态振型展开，具体步骤如下：将分布随机动载荷f(x,t,θ)表示为其分布函数T(x)与随机 时间历程P(t,θ)的乘积，如下式所示：S31: The random dynamic load distribution is independent of its random characteristics and time history, and its distribution function and random time history are all developed by using the mode shape. The specific steps are as follows: the distributed random dynamic load f(x, t, θ) is expressed. The product of its distribution function T(x) and the random time history P(t, θ) is as follows:f(x,t,θ)＝T(x)·P(t,θ) (3)，f(x,t,θ)=T(x)·P(t,θ) (3),将分布函数T(x)利用模态振型展开：The distribution function T(x) is expanded using the modal shape:
将式(4)代入式(3)，可得：Substituting equation (4) into equation (3), you can get:
其中a k(t,θ)＝d kP(t,θ)。 Where a k (t, θ) = d k P(t, θ).第i阶模态空间内的随机动载荷f i(t,θ)可以表示为： The random dynamic load f i (t, θ) in the i-th modal space can be expressed as:
其中L表示梁的长度；Where L represents the length of the beam;S32：利用模态振型函数的正交性，建立模态空间内随机动响应与随机动载荷之间的关系。具体步骤如下：S32: Using the orthogonality of the mode shape function, establishing a relationship between the random dynamic response and the random dynamic load in the modal space. Specific steps are as follows:结构在模态空间内的动力学方程为：The dynamic equations of the structure in the modal space are:
其中q i(t,θ)，和
分别为模态空间内的结构随机位移，速度和加速度，ζ i和m i分别为第i阶模态阻尼比和模态质量。将式(6)代入式(7)，利用结构模态振型的正交性条件，可以获得模态空间内随机动响应与随机动载荷之间的关系式，如下： Where q i (t, θ),
withThe random displacement, velocity and acceleration of the structure in the modal space, ζ i and m i are the i-th modal damping ratio and modal mass, respectively. Substituting equation (6) into equation (7), using the orthogonality condition of the structural mode shape, the relationship between the random dynamic response and the random dynamic load in the modal space can be obtained as follows:
- 根据权利要求1所述的随机动载荷空间分布及统计特征的识别方法，其特征在于，步骤S4中所述的在模态空间内，由随机动响应求解随机动载荷，具体包括以下步骤：The method for identifying a spatial distribution of random dynamic loads and a statistical feature according to claim 1, wherein in the modal space, the random dynamic response is solved by the random dynamic response in the modal space, which specifically includes the following steps:S41：根据结构位移，速度和加速度之间的导数关系，由第r次测量获得的结构模态位移q i,r(t)，对时间求导获取对应的模态速度和模态加速度；若测量获得的是结构加速度信号，同样可采用积分的方式获取速度和位移； S41: According to the structural displacement, the derivative relationship between the velocity and the acceleration, the structural modal displacement q i,r (t) obtained by the rth measurement, and the time derivative to obtain the corresponding modal velocity and modal acceleration; The measurement obtains the structural acceleration signal, and the speed and displacement can also be obtained by integrating;S42：由式(8)中的模态空间内随机位移q i(t,θ)的样本q i,r(t)及其导数，求解得到模态空间内随机动载荷a i(t,θ)的样本a i,r(t)。 S42: Calculating the random dynamic load a i (t, θ in the modal space by the sample q i,r (t) and its derivative of the random displacement q i (t, θ) in the modal space in the equation (8) The sample a i,r (t).
- 根据权利要求1所述的随机动载荷空间分布及统计特征的识别方法，其特征在于，步骤S5中所述的由模态空间内随机动载荷样本及振型函数求解结构上随机动载荷的空间分布和统计特征，具体包括以下步骤：The method for identifying a random dynamic load spatial distribution and a statistical feature according to claim 1, wherein the random dynamic load sample and the mode function in the modal space are used to solve the space of the random dynamic load on the structure described in step S5. Distribution and statistical characteristics, including the following steps:S51：随机动载荷的空间分布可以由下式计算：S51: The spatial distribution of the random dynamic load can be calculated by:
其中t 1为任意时刻； Where t 1 is any time;S52：随机动载荷的均值μ f(x,t)可以由下式计算： S52: The mean value of the random dynamic load μ f (x, t) can be calculated by:
S53：随机动载荷的方差Var f(x,t)可以由下式计算： S53: The variance of the random dynamic load Var f (x, t) can be calculated by:
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201710476795.7 | 2017-06-21 | ||
| CN201710476795.7A CN107423487B (en) | 2017-06-21 | 2017-06-21 | A kind of recognition methods of Random dynamic loads spatial distribution and statistical nature |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| WO2018233359A1 true WO2018233359A1 (en) | 2018-12-27 |
Family
ID=60426599
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| PCT/CN2018/083275 WO2018233359A1 (en) | 2017-06-21 | 2018-04-17 | Method for recognizing spatial distribution and statistical feature of random dynamic load |
Country Status (2)
| Country | Link |
|---|---|
| CN (1) | CN107423487B (en) |
| WO (1) | WO2018233359A1 (en) |
Families Citing this family (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107423487B (en) * | 2017-06-21 | 2018-06-12 | 东南大学 | A kind of recognition methods of Random dynamic loads spatial distribution and statistical nature |
| CN108038315B (en) * | 2017-12-15 | 2019-04-09 | 东南大学 | A kind of Random dynamic loads recognition methods based on spectrum stochastic finite meta-model |
| CN108491578B (en) * | 2018-02-13 | 2019-03-12 | 东南大学 | A kind of Random dynamic loads recognition methods based on perturbative matrix |
| CN110083886A (en) * | 2019-04-08 | 2019-08-02 | 西北工业大学 | A kind of helicopter vibration load simulating analysis based on finite element |
| CN113392547B (en) * | 2021-05-31 | 2024-01-05 | 东南大学 | Frequency domain distribution random dynamic load identification method considering spatial correlation |
| CN113392545B (en) * | 2021-05-31 | 2024-01-30 | 东南大学 | Time domain distribution random dynamic load identification method considering spatial correlation |
Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN102129520A (en) * | 2011-03-16 | 2011-07-20 | 哈尔滨工程大学 | Self-adaptive time domain identification method of dynamic load |
| US8577628B2 (en) * | 2009-04-10 | 2013-11-05 | University Of South Carolina | System and method for modal identification using smart mobile sensors |
| CN104090977A (en) * | 2014-07-22 | 2014-10-08 | 东南大学 | Random recognition method for bridge floor moving vehicle loads |
| CN104123463A (en) * | 2014-07-22 | 2014-10-29 | 东南大学 | Time domain identification method of random dynamic loads |
| CN107423487A (en) * | 2017-06-21 | 2017-12-01 | 东南大学 | A kind of recognition methods of Random dynamic loads spatial distribution and statistical nature |
Family Cites Families (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| GB201104413D0 (en) * | 2011-03-16 | 2011-04-27 | Airbus Operations Ltd | Stresses induced by random loading |
-
2017
- 2017-06-21 CN CN201710476795.7A patent/CN107423487B/en active Active
-
2018
- 2018-04-17 WO PCT/CN2018/083275 patent/WO2018233359A1/en active Application Filing
Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US8577628B2 (en) * | 2009-04-10 | 2013-11-05 | University Of South Carolina | System and method for modal identification using smart mobile sensors |
| CN102129520A (en) * | 2011-03-16 | 2011-07-20 | 哈尔滨工程大学 | Self-adaptive time domain identification method of dynamic load |
| CN104090977A (en) * | 2014-07-22 | 2014-10-08 | 东南大学 | Random recognition method for bridge floor moving vehicle loads |
| CN104123463A (en) * | 2014-07-22 | 2014-10-29 | 东南大学 | Time domain identification method of random dynamic loads |
| CN107423487A (en) * | 2017-06-21 | 2017-12-01 | 东南大学 | A kind of recognition methods of Random dynamic loads spatial distribution and statistical nature |
Also Published As
| Publication number | Publication date |
|---|---|
| CN107423487A (en) | 2017-12-01 |
| CN107423487B (en) | 2018-06-12 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| WO2018233359A1 (en) | Method for recognizing spatial distribution and statistical feature of random dynamic load | |
| US10228278B2 (en) | Determining a health condition of a structure | |
| WO2018233360A1 (en) | Distributed random dynamic load recognition method based on kl expansion | |
| WO2019011025A1 (en) | Finite element based method for identifying distributed random dynamic load | |
| Neu et al. | Fully automated operational modal analysis using multi-stage clustering | |
| US20160109323A1 (en) | Method for analyzing measured signal in resonance fatigue test and apparatus using the same | |
| Kim et al. | Automated peak picking using region‐based convolutional neural network for operational modal analysis | |
| Yao et al. | Blind modal identification using limited sensors through modified sparse component analysis by time‐frequency method | |
| Hu et al. | Local damage detection of membranes based on Bayesian operational modal analysis and three-dimensional digital image correlation | |
| Zhi et al. | Identification of wind loads on super-tall buildings by Kalman filter | |
| Le et al. | Computationally efficient stochastic approach for the fragility analysis of vertical structures subjected to thunderstorm downburst winds | |
| Wang et al. | Non-contact identification of rotating blade vibration | |
| Khoshnoudian et al. | A new damage index using FRF data, 2D-PCA method and pattern recognition techniques | |
| Wang et al. | Substructural identification of jack-up platform in time and frequency domains | |
| Chen et al. | Gradient-based point tracking method and its application in the modal test of a solar array model | |
| Rinn et al. | Stochastic method for in-situ damage analysis | |
| CN110411387B (en) | Frequency-based beam type structure additional mass preliminary positioning method | |
| Allada et al. | An unsupervised learning algorithm for computer vision-based blind modal parameters identification of output-only structures from video measurements | |
| Asadollahi et al. | Statistical analysis of modal properties of a cable-stayed bridge through long-term structural health monitoring with wireless smart sensor networks | |
| Dan et al. | Damping estimation by 2D dr_APES and its application to a real cable-stayed bridge | |
| Hsiung et al. | Analyzing the dynamic response of rotating blades in small-scale wind turbines | |
| Hoell et al. | Structural damage detection in wind turbine blades based on time series representations of dynamic responses | |
| Chu et al. | Bayesian spectral density approach for identification and uncertainty quantification of bridge section's flutter derivatives operated in turbulent flow | |
| Li et al. | Optimal sensor placement for structural health monitoring based on KL divergence | |
| Ge et al. | Structural damage detection approach using input-output monitoring system and transfer learning |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| 121 | Ep: the epo has been informed by wipo that ep was designated in this application |
Ref document number: 18819855 Country of ref document: EP Kind code of ref document: A1 |
|
| NENP | Non-entry into the national phase |
Ref country code: DE |
|
| 122 | Ep: pct application non-entry in european phase |
Ref document number: 18819855 Country of ref document: EP Kind code of ref document: A1 |
|
| 32PN | Ep: public notification in the ep bulletin as address of the adressee cannot be established |
Free format text: NOTING OF LOSS OF RIGHTS PURSUANT TO RULE 112(1) EPC (EPO FORM 1205A DATED 08.09.2020) |
|
| 122 | Ep: pct application non-entry in european phase |
Ref document number: 18819855 Country of ref document: EP Kind code of ref document: A1 |